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Mean field limit of a dynamical model of polymer systems

Science China mathematics 2013年第12期56卷 2591-2598页

作者： E Weinan SHEN Hao School of Mathematical Sciences and BICMR Peking University Department of Mathematics and PACM Princeton UniversityPrinceton NJ 08544 USA Program in Applied and Computational Mathematics Princeton UniversityPrinceton NJ 08544 USA

This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. Every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations （SPDEs）. We show that the mean field limit as N -＋ c~ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDE system of the polymers and the mean field limit equation are well-posed.

关键词： polymers mean field limit stochastic PDEs McKean-Vlasov equation

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Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime

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Communications in computational Physics 2011年第3期9卷 668-687页

作者： Shi Jin Hao Wu Xu Yang Department of Mathematics University of WisconsinMadisonWI 53706USA Department of Mathematical Sciences Tsinghua UniversityBeijing 10084China Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA

A novel Eulerian Gaussian beam method was developed in[8]to compute the Schrödinger equation efficiently in the semiclassical *** this paper,we introduce an efficient semi-Eulerian implementation of this *** new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing *** difference lies in that,we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these *** yields effectively a local level set implementation,and the beam summation can be carried out on the initial physical space instead of the phase *** a consequence,it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam ***,the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods,which is the topic of the second part of this *** numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.

关键词： Schrödinger equation semi-Eulerian Gaussian beam method high order methods

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Parallel spectral-element-Fourier simulation of turbulent flow over riblet-mounted surfaces

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Theoretical and computational Fluid Dynamics 1992年第4期3卷 219-229页

作者： Chu, Douglas Henderson, Ron Karniadakis, George Em Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics Princeton University Princeton 08544 NJ United States

The flow in a channel with its lower wall mounted with streamwise V-shaped riblets is simulated using a highly efficient spectral-element-Fourier method. The range of Reynolds numbers investigated is 500 to 4000, which corresponds to laminar, transitional, and turbulent flow states. Our results suggest that in the laminar regime there is no drag reduction, while in the transitional and turbulent regimes drag reduction up to 10% exists for the riblet-mounted wall in comparison with the smooth wall of the channel. For the first time, we present detailed turbulent statistics in a complex geometry. These results are in good agreement with available experimental data and provide a quantitative picture of the drag-reduction mechanism of the riblets. © 1992 Springer-Verlag.

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Numerical implicitization

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Journal of Software for Algebra and Geometry 2019年第1期9卷 55-63页

作者： Chen, Justin Kileel, Joe School of Mathematics Georgia Institute of Technology Atlanta GA United States Program in Applied and Computational Mathematics Princeton University Princeton NJ United States

We present the NumericalImplicitization.m2 package for Macaulay2, which allows for user-friendly computation of the invariants of the image of a polynomial map, such as dimension, degree, and Hilbert function values. This package relies on methods of numerical algebraic geometry, including homotopy continuation and monodromy. © 2019, Mathematical Sciences Publishers. All rights reserved.

关键词： homotopy continuation implicitization interpolation monodromy numerical algebraic geometry

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An Ant Colony Algorithm Approach to the Vehicle Routing Problem and Variants

An Ant Colony Algorithm Approach to the Vehicle Routing Prob...

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2024 Systems and Information Engineering Design Symposium, SIEDS 2024

作者： Tablada, Tulio Davis, Cleon Palumbo, Neil Johns Hopkins University Electrical and Computer Engineering Program Johns Hopkins University Applied and Computational Mathematics Program United States

ISBN: (纸本)9798350385144

In the basic vehicle routing problem (VRP), a vehicle must deliver goods from one centralized warehouse to multiple customers efficiently. Several VRP variants and constraints exist, including different product types, specific delivery times, multiple warehouses, vehicle fuel constraints, and pick-up from one location and delivery to another. This work proposes and demonstrates a flexible algorithm for solving the VRP via ant colony optimization (ACO) that can address many of the variants discussed above. ACO algorithms mimic the behavior of ants, learning optimal paths to a food source and back to the nest based on stigmergic behavior. This work compares a proposed, more flexible ACO algorithm to a traditional optimization algorithm that is implemented in the Google VRP solver. Readily available data were used to test and demonstrate results. Several VRP variants were implemented in both the (proposed) flexible ACO algorithm and the Google VRP solver. The flexible ACO algorithm performed better in terms of distance traveled versus the Google VRP solver for two variants and worse for three other cases. However, the Google VRP solver was not able to solve some of the VRP variant combinations considered here and failed when solving some backhaul datasets. Because the flexible ACO algorithm was able to better handle many case variations, it may be an attractive alternative optimization tool. © 2024 IEEE.

关键词： Ant colony optimization

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From Dyson models to many-body quantum chaos

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Physical Review B 2025年第3期111卷 035147-035147页

作者： Alexei Andreanov Matteo Carrega Jeff Murugan Jan Olle Dario Rosa Ruth Shir Center for Theoretical Physics of Complex Systems Institute for Basic Science (IBS) Daejeon 34126 Korea Basic Science Program Korea University of Science and Technology (UST) Daejeon 34113 Korea CNR-SPIN Via Dodecaneso 33 16146 Genova Italy The Laboratory for Quantum Gravity & Strings Department of Mathematics and Applied Mathematics University of Cape Town Cape Town 7701 South Africa The National Institute for Theoretical and Computational Sciences Private Bag X1 Matieland 7602 South Africa Max Planck Institute for the Science of Light 91058 Erlangen Germany ICTP South American Institute for Fundamental Research Instituto de Física Teórica UNESP—University of Estadual Paulista Rua Dr. Bento Teobaldo Ferraz 271 01140-070 São Paulo SP Brazil Department of Physics and Materials Science University of Luxembourg L-1511 Luxembourg Luxembourg

A deep understanding of the mechanisms underlying many-body quantum chaos is one of the big challenges in contemporary theoretical physics. We tackle this problem in the context of a set of perturbed quadratic Sachdev-Ye-Kitaev (SYK) Hamiltonians defined on graphs. This allows us to disentangle the geometrical properties of the underlying single-particle problem and the importance of the interaction terms, showing that the former is the dominant feature ensuring the single-particle to many-body chaotic transition. Our results are verified numerically with state-of-the-art numerical techniques, capable of extracting eigenvalues in a desired energy window of very large Hamiltonians. Our approach essentially provides a new way of viewing many-body chaos from a single-particle perspective.

关键词： Quantum chaos

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NUMERICAL STUDY OF A 2-POINT CORRELATION-FUNCTION AND LIOUVILLE FIELD PROPERTIES IN 2-DIMENSIONAL QUANTUM-GRAVITY

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MODERN PHYSICS LETTERS A 1991年第12期6卷 1115-1131页

作者： AGISHTEIN, ME BENAV, R MIGDAL, AA SOLOMON, S Program in Applied and Computational Mathematics Fine Hall Princeton University Princeton NJ 08544–1000 USA Nuclear Physics Department Weizmann Institute of Science Rehovot 76100 Israel Physics Department and Program in Applied and Computational Mathematics Fine Hall Princeton University Princeton NJ 08544–1000 USA Physics Department Hebrew University Jerusalem Israel

Two-point Green's function is measured on the manifolds of a 2-dimensional quantum gravity. The recursive sampling technique is used to generate the triangulations, lattice sizes being up to hundred thousand triangles. The grid Laplacian was inverted by means of the algebraic multi-grid solver. The free field model of the Quantum Gravity assumes the Gaussian behavior of Liouville field and curvature. We measured histograms as well as six momenta of these fields. Our results support the Gaussian assumption.

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Hybrid spectral-element-low-order methods for incompressible flows

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Journal of Scientific Computing 1991年第2期6卷 79-100页

作者： Henderson, Ron Karniadakis, George Em. Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics Princeton University Princeton 08544 New Jersey United States

In this article we present a new formulation for coupling spectral element discretizations to finite difference and finite element discretizations addressing flow problems in very complicated geometries. A general iterative relaxation procedure (Zanolli patching) is employed that enforces C¹ continuity along the patching interface between the two differently discretized subdomains. In fluid flow simulations of transitional and turbulent flows the high-order discretization (spectral element) is used in the outer part of the domain where the Reynolds number is effectively very high. Near "rough" wall boundaries (where the flow is effectively very viscous) the use of low-order discretizations provides sufficient accuracy and allows for efficient treatment of the complex geometry. An analysis of the patching procedure is presented for elliptic problems, and extensions to incompressible Navier-Stokes equations are implemented using an efficient high-order splitting scheme. Several examples are given for elliptic and flow model problems and performance is measured on both serial and parallel processors. © 1991 Plenum Publishing Corporation.

关键词： complex geometries finite differences finite elements Incompressible flows spectral elements

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Numerical analysis of dynamical systems

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Acta Numerica 1994年 3卷 467-572页

作者： Stuart, Andrew M. Program in Scientific Computing and Computational Mathematics Division of Applied Mechanics Stanford University California CA 94305-4040 United States

This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long-time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error constant typically grows like the exponential of the time interval under consideration. Instead of comparing trajectories, the effect of discretization on various sets which are invariant under the evolution of the underlying differential equation is studied. Such invariant sets are crucial in determining long-time dynamics. The particular invariant sets which are studied are equilibrium points, together with their unstable manifolds and local phase portraits, periodic solutions, quasi-periodic solutions and strange attractors. Particular attention is paid to the development of a unified theory and to the development of an existence theory for invariant sets of the underlying differential equation which may be used directly to construct an analogous existence theory (and hence a simple approximation theory) for the numerical method. © 1994, Cambridge University Press. All rights reserved.

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ON THE SOLUTION OF LAPLACE’S EQUATION IN THE VICINITY OF TRIPLE JUNCTIONS

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Pure and applied Analysis 2020年第2期2卷 447-476页

作者： Hoskins, Jeremy Rachh, Manas Applied Mathematics Program Yale University New Haven CT United States Center for Computational Mathematics Flatiron Institute New York NY United States

We characterize the behavior of solutions to systems of boundary integral equations associated with Laplace transmission problems in composite media consisting of regions with polygonal boundaries. In particular we consider triple junctions, i.e., points at which three distinct media meet. We show that, under suitable conditions, solutions to the boundary integral equations in the vicinity of a triple junction are well-approximated by linear combinations of functions of the form t^β, where t is the distance of the point from the junction and the powers β depend only on the material properties of the media and the angles at which their boundaries meet. Moreover, we use this analysis to design efficient discretizations of boundary integral equations for Laplace transmission problems in regions with triple junctions and demonstrate the accuracy and efficiency of this algorithm with a number of examples. © 2020, Mathematical Sciences Publishers. All rights reserved.

关键词： boundary integral equations corners multiple junction interfaces potential theory singular solutions

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